Space of modular forms of level 240, weight 10, and trivial character

• Label: 240.10.a
• Level: $240$
• Weight: $10$
• Character orbit: 240.a
• Rep. character: $\chi_{240}(1,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $22$
• Sturm bound: $480$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	240.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(480\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(240))\).

	Total	New	Old
Modular forms	444	36	408
Cusp forms	420	36	384
Eisenstein series	24	0	24

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(3\)	\(5\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(5\)
\(+\)	\(+\)	\(-\)	$-$	\(5\)
\(+\)	\(-\)	\(+\)	$-$	\(4\)
\(+\)	\(-\)	\(-\)	$+$	\(4\)
\(-\)	\(+\)	\(+\)	$-$	\(5\)
\(-\)	\(+\)	\(-\)	$+$	\(4\)
\(-\)	\(-\)	\(+\)	$+$	\(4\)
\(-\)	\(-\)	\(-\)	$-$	\(5\)
Plus space			\(+\)	\(17\)
Minus space			\(-\)	\(19\)

Trace form

\( 36 q - 162 q^{3} + 7552 q^{7} + 236196 q^{9} + O(q^{10}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(240))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs			Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	5
240.10.a.a	$240$	$10$	240.a	1.a	$1$	$1$	$1$	$123.609$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-81\)	\(-625\)	\(-7196\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{4}q^{3}-5^{4}q^{5}-7196q^{7}+3^{8}q^{9}+\cdots\)
240.10.a.b	$240$	$10$	240.a	1.a	$1$	$1$	$1$	$123.609$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-81\)	\(625\)	\(-2408\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{4}q^{3}+5^{4}q^{5}-2408q^{7}+3^{8}q^{9}+\cdots\)
240.10.a.c	$240$	$10$	240.a	1.a	$1$	$1$	$1$	$123.609$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-81\)	\(625\)	\(7680\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{4}q^{3}+5^{4}q^{5}+7680q^{7}+3^{8}q^{9}+\cdots\)
240.10.a.d	$240$	$10$	240.a	1.a	$1$	$1$	$1$	$123.609$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(81\)	\(-625\)	\(-6332\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{4}q^{3}-5^{4}q^{5}-6332q^{7}+3^{8}q^{9}+\cdots\)
240.10.a.e	$240$	$10$	240.a	1.a	$1$	$1$	$1$	$123.609$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(81\)	\(-625\)	\(-3836\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{4}q^{3}-5^{4}q^{5}-3836q^{7}+3^{8}q^{9}+\cdots\)
240.10.a.f	$240$	$10$	240.a	1.a	$1$	$1$	$1$	$123.609$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(81\)	\(-625\)	\(-3164\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{4}q^{3}-5^{4}q^{5}-3164q^{7}+3^{8}q^{9}+\cdots\)
240.10.a.g	$240$	$10$	240.a	1.a	$1$	$1$	$1$	$123.609$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(81\)	\(-625\)	\(5988\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{4}q^{3}-5^{4}q^{5}+5988q^{7}+3^{8}q^{9}+\cdots\)
240.10.a.h	$240$	$10$	240.a	1.a	$1$	$1$	$1$	$123.609$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(81\)	\(625\)	\(2464\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3^{4}q^{3}+5^{4}q^{5}+2464q^{7}+3^{8}q^{9}+\cdots\)
240.10.a.i	$240$	$10$	240.a	1.a	$1$	$1$	$1$	$123.609$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(81\)	\(625\)	\(7168\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3^{4}q^{3}+5^{4}q^{5}+7168q^{7}+3^{8}q^{9}+\cdots\)
240.10.a.j	$240$	$10$	240.a	1.a	$1$	$1$	$1$	$123.609$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(81\)	\(625\)	\(10336\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3^{4}q^{3}+5^{4}q^{5}+10336q^{7}+3^{8}q^{9}+\cdots\)
240.10.a.k	$240$	$10$	240.a	1.a	$1$	$2$	$2$	$123.609$	\(\Q(\sqrt{161}) \)	None	✓	$1$	$1$	\(0\)	\(-162\)	\(-1250\)	\(-3136\)		$+$	$+$	$+$	$+$	$2^{3}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q-3^{4}q^{3}-5^{4}q^{5}+(-1568-3\beta )q^{7}+\cdots\)
240.10.a.l	$240$	$10$	240.a	1.a	$1$	$2$	$2$	$123.609$	\(\Q(\sqrt{457}) \)	None	✓	$1$	$0$	\(0\)	\(-162\)	\(-1250\)	\(-928\)		$-$	$+$	$+$	$-$	$2^{3}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q-3^{4}q^{3}-5^{4}q^{5}+(-464-\beta )q^{7}+\cdots\)
240.10.a.m	$240$	$10$	240.a	1.a	$1$	$2$	$2$	$123.609$	\(\Q(\sqrt{4729}) \)	None	✓	$1$	$0$	\(0\)	\(-162\)	\(-1250\)	\(11872\)		$-$	$+$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-3^{4}q^{3}-5^{4}q^{5}+(5936-7\beta )q^{7}+\cdots\)
240.10.a.n	$240$	$10$	240.a	1.a	$1$	$2$	$2$	$123.609$	\(\Q(\sqrt{22057}) \)	None	✓	$1$	$1$	\(0\)	\(-162\)	\(1250\)	\(2072\)		$-$	$+$	$-$	$+$	$2^{3}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q-3^{4}q^{3}+5^{4}q^{5}+(1036-\beta )q^{7}+3^{8}q^{9}+\cdots\)
240.10.a.o	$240$	$10$	240.a	1.a	$1$	$2$	$2$	$123.609$	\(\Q(\sqrt{193}) \)	None	✓	$1$	$0$	\(0\)	\(-162\)	\(1250\)	\(4280\)		$+$	$+$	$-$	$-$	$2^{3}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q-3^{4}q^{3}+5^{4}q^{5}+(2140+9\beta )q^{7}+\cdots\)
240.10.a.p	$240$	$10$	240.a	1.a	$1$	$2$	$2$	$123.609$	\(\Q(\sqrt{46}) \)	None	✓	$1$	$0$	\(0\)	\(162\)	\(-1250\)	\(4424\)		$+$	$-$	$+$	$-$	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+3^{4}q^{3}-5^{4}q^{5}+(2212+7\beta )q^{7}+\cdots\)
240.10.a.q	$240$	$10$	240.a	1.a	$1$	$2$	$2$	$123.609$	\(\Q(\sqrt{29446}) \)	None	✓	$1$	$0$	\(0\)	\(162\)	\(-1250\)	\(4808\)		$+$	$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+3^{4}q^{3}-5^{4}q^{5}+(2404+\beta )q^{7}+3^{8}q^{9}+\cdots\)
240.10.a.r	$240$	$10$	240.a	1.a	$1$	$2$	$2$	$123.609$	\(\Q(\sqrt{241}) \)	None	✓	$1$	$0$	\(0\)	\(162\)	\(1250\)	\(-14112\)		$-$	$-$	$-$	$-$	$2^{5}\cdot 3$	$\mathrm{SU}(2)$	\(q+3^{4}q^{3}+5^{4}q^{5}+(-84^{2}-7\beta )q^{7}+\cdots\)
240.10.a.s	$240$	$10$	240.a	1.a	$1$	$2$	$2$	$123.609$	\(\Q(\sqrt{46}) \)	None	✓	$1$	$1$	\(0\)	\(162\)	\(1250\)	\(-2176\)		$+$	$-$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+3^{4}q^{3}+5^{4}q^{5}+(-1088+13\beta )q^{7}+\cdots\)
240.10.a.t	$240$	$10$	240.a	1.a	$1$	$2$	$2$	$123.609$	\(\Q(\sqrt{406}) \)	None	✓	$1$	$1$	\(0\)	\(162\)	\(1250\)	\(-1792\)		$+$	$-$	$-$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+3^{4}q^{3}+5^{4}q^{5}+(-896+7\beta )q^{7}+\cdots\)
240.10.a.u	$240$	$10$	240.a	1.a	$1$	$3$	$3$	$123.609$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-243\)	\(-1875\)	\(1276\)		$+$	$+$	$+$	$+$	$2^{10}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q-3^{4}q^{3}-5^{4}q^{5}+(425+\beta _{1})q^{7}+3^{8}q^{9}+\cdots\)
240.10.a.v	$240$	$10$	240.a	1.a	$1$	$3$	$3$	$123.609$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-243\)	\(1875\)	\(-9736\)		$+$	$+$	$-$	$-$	$2^{10}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q-3^{4}q^{3}+5^{4}q^{5}+(-3245+\beta _{1})q^{7}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(240))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(240)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 16}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 10}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 12}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 10}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 5}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)